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Autoencoders and latent space fragmentation – X – a method to create suitable latent vectors for the generation of human face images

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My present post series explores options to use a standard convolutional Autoencoder [AE] for the creation of images with human faces. The face generation should based on random input to the AE’s Decoder. On our quest for a suitable method we have meanwhile learned a lot about other aspects of Autoencoders, vector distributions in multi-dimensional latent spaces and generative methods for our special case:

  • Methods to create statistical latent vectors [z-vectors] as input for the AE’s Decoder must be chosen carefully. Among other things: It is difficult to create a bunch of random vectors which cover wider areas in the vastness of a multidimensional space. So the z-vector creation must be adjusted to specific requirements.
  • After having been trained with CelebA images a convolutional AE fills a limited and coherent region in the latent space with z-points for the training images. This latent space region appears to be critical for successful image creation: Statistically generated z-vectors should point to this region. The core of the z-point distribution gets filled relatively densely.
  • A convolutional AE maps human face images onto an approximate multivariate normal distribution. This gives the inner core of the z-point distribution the structure of a multidimensional ellipsoid. The projections of this ellipsoid onto 2-dimensional coordinate planes show characteristic nested elliptic contour lines.
  • As the main axes of these ellipses were inclined with different angle towards the axes of chosen coordinate planes we concluded that linear correlations mark average dependencies between the z-vector components. Limiting conditions imposed by these correlations must also be fulfilled by z-vectors used as the Decoder’s input.

See previous posts in this series for more details. In particular, the last 2 posts

Autoencoders and latent space fragmentation – IX – PCA transformation of the z-point distribution for CelebA

Autoencoders and latent space fragmentation – VIII – approximation of the latent vector distribution by a multivariate normal distribution and ellipses

have shown that the density distribution for the z-points really exhibits elliptic contour lines in the original coordinate system of the latent space and (!) in the target coordinate system of a PCA transformation.

In this post we use our gathered knowledge: I present a first simple method to generate z-vectors which point to the latent space region filled by z-points for CelebA images. These z-vectors will fulfill the general and limiting elliptic conditions for their components.

Decomposing the full problem of latent vector generation into a sequence of 2-dimensional problems

The nice thing about multivariate Gaussian distributions with linear correlations between the vector components is the following: We can reduce the problem of choosing proper component values to a series of 2-dimensional restrictions. Firstly we can use characteristic properties of the Gaussian distribution for each component. And secondly we can use confidence ellipses in 2-dimensional coordinate planes to restrict the component values to allowed intervals.

Ellipses are most easy to handle when their axes are aligned with the axes of the coordinate system in which we describe them. So, let us assume that we know an affine transformation T to a new coordinate system which also has orthogonal axes and supports the following special transformation properties for a multivariate normal density distribution:

  1. T maps nested elliptic contour lines of the multidimensional density distribution and in particular confidence ellipses for component pairs in the original coordinate system to nested elliptic contours and confidence ellipses in the new coordinate system.
  2. Taligns the centers of the transformed ellipses with the origin of the new coordinate system.
  3. T aligns the main axes of the mapped ellipses with the axes of the new coordinate system.
  4. T is reversible.

How could we then use the transformed data for vector-creation?

In the new coordinate system, a contour ellipse in a chosen coordinate plane for the axes-indices (i, j) may have main diameters of size

d1 = 2 * a    and    d2 = 2 * b.

We then can first select a random v_i value to fall into a range [-a * fact, a * fact].

– fact * a    <    v_i    <    fact * a

With fact being a proper factor. This factor defines a confidence level in the new coordinate system. With the value of v_i fixed and b being the half-diameter in the orthogonal direction the correlation condition for the z-point distribution says that the v_j value must fall into an interval [-c, c] defined by:

-c    <    v_j    <    c,
with c = b * fact * sqrt(1 – x**2 / (fact * a)**2)

But within these limits we can again choose the v_j-value freely. Below I use a simple random-function for a constant probability density to pick a value.

However: It would not be enough to restrict the coordinates to the conditions of just one ellipse! The components of the created vectors must in parallel fulfill elliptic conditions for all of the possible pairs of vector-components. I.e. we may need to adapt the v_j values gained from the analysis of a fist 2D-ellipse to further conditions of other ellipses and component pairs. This can be achieved by an iteration. For z_dim = 256 this involves a total of 32640 checks and possible value-adaptions to each and all of the allowed value ranges.

In addition: The order by which the component-pairs and their conditions are investigated must be randomized to get real statistical vector distributions.

Eventually the resulting vector components must be re-transformed into the original coordinate system of the latent space.

The ellipse for the “core’s boundary” in the original coordinate system will be defined by the chosen confidence level of the ellipsoidal normal distribution. We saw already that a confidence level of σ = 2.0 defines the transition to outer regions of the z-point density distribution quite well.

This all sounds manageable by relative simple Python programs. But: Do we know a proper transformation T? Yes, we do: A PCA-transformation of the z-point density distribution has all the properties discussed above.

Using half maximum values after a PCA transformation of the z-point distribution

The last post proved that a PCA transformation maps ellipses onto ellipses for component pairs in the transformed PCA coordinate system. The advantage of the ellipses there is that their main axes are on average well aligned with the orthogonal PCA coordinate axes. Gaussians for the number density distribution per component are mapped to Gaussians for the new components in the transformed coordinate system. So, the basic idea for a proper z-vector generation is:

  1. Take the multivariate normal z-point distribution for the training images in the AE’s latent space.
  2. Apply a PCA analysis to diagonalize the correlation matrix and transform the z-vector components to the PCA coordinate system.
  3. Use the ellipses in coordinate planes of the PCA coordinate system to create random z-vector components fulfilling all required conditions there.
  4. Re-transform the resulting z-vector components into the original coordinate system of the latent space.

Point 3 in our method is covered by a numerical analysis of the Gaussians in the PCA-coordinate system. We determine the half-width numerically by analyzing the density distribution with the help of sampling intervals. This simple method has resolution limits related to the size of the sampling interval. This has consequences for PCA components with a small standard deviation. We saw already in the last posts that such distributions appear for higher PCA components at the lower end of the explained variance.

Does the suggested method work?

The convolutional AE we work with was defined in previous posts with 4 Conv2D layers in the Encoder and 4 Conv2DTranspose layers in Decoder. The number of latent space dimensions was z_dim = 256. The AE network was trained on CelebA images. I do not want to bore you with details of the codes for the creation of z-vectors consistent to the resulting elliptic conditions. It is all standard. The PCA-transformation can e.g. be taken from the sklearn-package.

I have applied a constant probability density to choose a random value within the allowed ranges for the component values of the aspired z-vectors in the PCA coordinate system. For the plots below I have used the most important 50 to 105 PCA components (out of 256). The plots include confidence ellipses on a level of σ = 2.2. I derived the confidence ellipses by directly evaluating the standard deviations of the transformed distribution data in all coordinate directions.

The first plot shows you such an ellipse for the coordinate plane corresponding to the first two, most important PCA components. The orange points mark 20 z-points defined by 20 randomly z-vectors fulfilling all elliptic conditions. The plot contains 120,000 z-points for images out of the 170,000 CelebA pictures used during training.

Generated statistical vectors in the PCA coordinate system

For elliptic contour lines see the last post before the present one in this series. The next plot shows the same generated 20 z-vectors for other component-combinations among the first 20 of the most important PCA-components. The plots contain a selection of 60,000 z-points.

The outer z-points points do not always indicate that we have elliptic contours in the denser core of the displayed 2-dimensional distributions. But see the last post for proofs that the inner core inside the red ellipse really displays elliptic contours. You see that all random vectors lie within the 2-σ-ellipses.

The next plot shows the generated z-vectors in the original coordinate system of the latent space. The component values were back-transformed from the PCA-system to the original coordinate system.

Generated statistical z-vectors after an inverse PCA transformation to the original coordinate system of the latent space

We get similar plots for other component pairs. And of course for other generated vectors.

Generated statistical z-vectors in the PCA coordinate system

Generated statistical z-vectors after an inverse PCA transformation to the original coordinate system of the latent space

Technically we have obviously achieved what we wanted: Our generated statistical vectors are distributed within the core of our multidimensional ellipsoid.

Note that this method fortunately works even when we use a limited number of the PCA components, only. This is due to intricate properties of a PCA transformation which guarantee that a back-transformation puts the resulting points close to the original ones even when we omit less important PCA components. I cannot discuss the math-details in this blog. You have to see scientific literature for this. An introduction is e.g. provided by https://arxiv.org/pdf/1404.1100.pdf.

For me this property of the PCA transformation was helpful when I ran into the resolution problem for a proper half-width of the Gaussians. Taking 256 components lead to errors as elliptic conditions for very narrow Gaussians were not properly defined and some of the created vectors left the allowed value ranges.

Resulting face images

Let us look at some results. First I want to remind you from where we started:

Failed trials with improper random z-vectors based on constant probability densities

A simple random generator used in the beginning was totally inapt to feed the AE’s Decoder with proper statistical z-vectors. And now – look at the following plots. They were produced for a varying number of PCA components between 50 and 120, 100000 statistically selected z-points within a 3 σ-level for the PCA-transformation and various factors 0.6 < fact < 0.8 used upon a half-width corresponding to a confidence level of 2.35 σ:

In some cases – for a higher number of PCA components – we even see smaller details of the face images and a reasonable transition to some kind of hairdo. Please remember that z_dim = 256 is a pretty low number for the latent space to cover the encoding of face details. And celebrities as covered by CelebA use make-up ….

In case you think the above result is not noteworthy: Please remember that we talk about a simple standard Autoencoder and not about a Variational Autoencoder and neither about a transformer based Autoencoder. No fancy additions to cost functions or special layers. And who ever has read the very instructive book of D. Foster on “Generative Deep Learning” (1st edition, O’Reilly) may compare his images to mine. And I have used a lower resolution of the original images than D. Foster. Just to motivate people to look a bit deeper into properties of data distributions in latent spaces.

Conclusion and outlook

We have come a lot closer to our objective of using a standard minimal Autoencoder for generative purposes. On our way, we got a much deeper understanding of the vector-distribution a trained AE creates in its latent space for human face images.

The method presented in this post to create reasonable statistical z-vectors still has its limits and there is a lot of open space for improvements. Attentive readers may e.g. ask: Why did he not use confidence ellipses directly? And why not the ellipses found in the original coordinate system of the latent space? And what about micro-correlations? And are there clusters for certain properties as the hair-color, sex, smiling, etc. in the multivariate z-point distribution in the AE’s latent space?

I will discuss these topics in further posts. In the meantime keep in mind that the basic point for turning a standard Autoencoder into a generative tool is to understand how it fills its latent space.

Note also that I myself have speculated in other posts of this blog that failures of using standard AEs for generative purposes may have their ultimate reason in the micro-structure of the z-point distribution. The present results render these previous ideas of mine plain wrong.

Links to previous posts of this series

Autoencoders and latent space fragmentation – IX – PCA transformation of the z-point distribution for CelebA

Autoencoders and latent space fragmentation – VIII – approximation of the latent vector distribution by a multivariate normal distribution and ellipses

Autoencoders and latent space fragmentation – VII – face images from statistical z-points within the latent space region of CelebA

Autoencoders and latent space fragmentation – VI – image creation from z-points along paths in selected coordinate planes of the latent space

Autoencoders and latent space fragmentation – V – reconstruction of human face images from simple statistical z-point-distributions?

Autoencoders and latent space fragmentation – IV – CelebA and statistical vector distributions in the surroundings of the latent space origin

Autoencoders and latent space fragmentation – III – correlations of latent vector components

Autoencoders and latent space fragmentation – II – number distributions of latent vector components

Autoencoders and latent space fragmentation – I – Encoder, Decoder, latent space

 

And before we forget it: Besides the Putler in the east there is also an extremist right-wing, semi-fascistic party in Germany on a record high support level in the population of 18%. This is a party which wants to stop all sanctions against the Russian aggressor in the ongoing war in Ukraine. You see the pattern behind this? This party is presently becoming bigger in number of supporters than the government leading social democrats. So, there is more at stake at present in Europe than the war in Ukraine. We need to defend our democracies with all the means of democracies. And its time to ask for more decisive legal action against a party which already is under observation of the German internal secret service.

 

Autoencoders and latent space fragmentation – VIII – approximation of the latent vector distribution by a multivariate normal distribution and ellipses

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This post series is about creative abilities of convolutional Autoencoders [AE] which have been trained on a set of human face images. The objectives of this series and its numerical experiments are:

  • We want to create images with human faces from statistical z-vectors and related z-points in the AE’s latent space [z-space or LS]. Image creation will be done with the help of the AE’s Decoder after a training on the CelebA dataset.
  • We work with a standard Autoencoder, only. I.e., we do NOT add any artificial layers and cost terms to the Autoencoder’s layer structure (as it is done e.g. in Variational Autoencoders).
  • We analyze the position, shape and internal structure of the multidimensional z-vector distribution created by the AE’s Encoder after training. We assume that generated statistical z-vectors must point to respective regions of the latent space to guarantee images with reasonable content.
  • We raise the question whether simple statistical generator algorithms are sufficient to cover these regions with statistical z-vectors.

Our numerical experiments gave us some indications that such an endeavor is indeed feasible. In addition the third objective may give us some insight into the rules a trained AE follows when it encodes information about human faces into vectors of its latent space.

We have already studied the “natural” z-vector distribution created by a convolutional Autoencoder for CelebA images after a thorough training. The related z-point distribution fortunately filled just one confined and coherent off-center region of the AE’s latent space. Our experiments have furthermore shown that we must indeed restrict the statistical z-vector creation such that the vectors point to this particular region. Otherwise we will not get reasonable images. For details see the previous posts.

Autoencoders and latent space fragmentation – VII – face images from statistical z-points within the latent space region of CelebA
Autoencoders and latent space fragmentation – VI – image creation from z-points along paths in selected coordinate planes of the latent space
Autoencoders and latent space fragmentation – V – reconstruction of human face images from simple statistical z-point-distributions?
Autoencoders and latent space fragmentation – IV – CelebA and statistical vector distributions in the surroundings of the latent space origin
Autoencoders and latent space fragmentation – III – correlations of latent vector components
Autoencoders and latent space fragmentation – II – number distributions of latent vector components
Autoencoders and latent space fragmentation – I – Encoder, Decoder, latent space

The frustrating point so far was that simple methods for creating statistical vectors fail to put the end-points of the z-vectors into the relevant latent space region. In particular methods based on constant probability distributions within a common value interval for all z-vector components are doomed to miss the interesting region due to intricate mathematical reasons.

Afterward we tried to restrict the component values of test vectors to intervals defined by the shape of the number distribution for the values of each component of the CelebA related z-vectors. Such a distribution is nothing else than a one-dimensional probability density function for our special set of encoded CelebA samples: The function describes the probability that a component of a z-vector for human face images gets a value within a certain small value range. The probability distributions for all z-vector components were bell shaped and showed clear transitions to flat wings with very low values. See the plots below. This allowed us to define a value range

d_j_l   <   x_j   <   d_j_h

for each vector component x_j.

But keeping statistical values per component within the identified respective interval was not a sufficient restriction. We saw this clearly in the last post from significant irregular fluctuations in the reconstructed images. Obviously the components of statistically generated z-vectors must in addition fulfill correlation conditions.

The questions which I want to answer in this post are:

  • Can we approximate the 1-dimensional probability density functions for the z-vector components by some simple and common mathematical function?
  • What kind of correlations do we find between the components of the z-vectors encoding the information of human face images?
  • Can we derive some mathematical description of the multivariate z-vector distribution created by convolutional AEs for human face images in the AE’s multidimensional latent space?

Correlations are to be expected …

Please note: We deal with a multidimensional problem. A single latent vector encodes information about a human face image via all of its component values and by relations between these values. Regarding the purpose and the task an AE has to fulfill, it would be naive to assume that the components of our multi-dimensional z-vectors were independently organized. A z-vector encodes information for a convolutional Decoder to combine patterns detected by the Encoder and represented in neural (feature) maps of the networks to create an image. This is a subtle business. Just think about what you do when you draw a sketch of a human face. There are a lot of rules you follow.

When you think about the properties of basic feature patterns in a human face you would certainly assume that the pixel data of a corresponding image show strong correlations. This is among other things due to obvious symmetries – not excluding fluctuations of basic parameters describing human face features. But a nose tends to be at a position below the eyes and at a mid-distance of the eyes. In additions fluctuations of face features would on average respect certain limits given by natural proportions of a face. It would therefore be unreasonable to assume that the input for a Decoder to create a superposition of elementary patterns consists of un-correlated data. Instead the patterns in the original data should not only lead to well adjusted weights in the convolutional networks’ feature maps, but also to well regulated structural elements in the data distribution in the target space of the information encoding, namely in the latent space.

If the relations of the vector components were of a complex, highly non-linear kind and involved many dimensions at the same time we might be lost. But the results we have gained so far indicate a proper common structure of at least the density function for the individual components. This gives us some hope that the multidimensional problem somehow involves well defined 1-dimensional constituents. Whether this a sign that the multidimensional structure of the z-vector distribution can be decomposed into low-dimensional relations remains to be seen.

Observations regarding the z-vector distribution created by convolutional Autoencoders for human face images

Coordinate values of the z-points are identical to z-vector component values when we fix the end of each vector to the origin of the latent space coordinate system. The z-vector distribution thus directly corresponds to a z-point density distribution in the orthogonal coordinate system of the AE’s multi-dimensional LS. We have already made three interesting observations regarding these distributions:

  • The individual probability density function for a selected component of the latent vectors has a bell-shaped form. One , therefore, is tempted to think of a Gaussian function. This would indicate a possible normal distribution for the coordinate values of the z-points along each of the selected coordinate axes.
    Note: This does not exclude that the probability distributions for the components are correlated in some complex way.
  • When we plotted the projection of the z-point distribution onto 2-dimensional coordinate planes (for selected pairs of coordinate axes) then almost all of the resulting 2-dimensional density distributions seemed to have a defined core with an ellipsoidal form of its boundary.
  • For certain component- or axis-pairs the main axes of the apparent ellipses for pair-wise density function appeared rotated against the coordinate axes. The elongated regular and more or less symmetric forms showed a diagonal orientation (with different angles). This alone signals a strong correlation between related two vector components. Indeed we found high values for certain elements of the matrix of normalized Pearson correlation coefficients for the multi-dimensional distribution of z-vector component values.

These observations are not unrelated; they indicate a clear pattern of dependencies and correlations of the distributions for the variables in place. Regarding the data basis we have to keep five things in mind:

  • We treat the z-point distribution for CelebA images as a multi-dimensional probability density distribution. During the analysis we look in particular at 2-dimensional projections of this distribution onto planes spanned by a selected pair of orthogonal axes of the LS coordinate system. We also consider the one-dimensional value distributions for z-vector components. In this sense we regard the z-vector components as logically separate variables.
  • The data used are numbers of z-points counted in finite 1d-intervals, 2d-rectangles or multidimensional cuboids. We fit idealized functions to the respective discrete bar plots. Even if there is a good 1d-fit fluctuations may especially get visible in multidimensional plots for correlated data. A related probability density requires a normalization. We drop the resulting constant factors in the qualitative discussions below.
  • Statistical (un-)correlation of statistical variable distributions must NOT to be confused with underlying variable (in-) dependency. Linear correlations can be reduced to zero by coordinate transformations without eliminating the original variable dependencies.
  • Pearson correlation coefficients are sensitive to linear elements in the relations of logically separate variable distributions. They can not fully cover non-linear distribution relations or covered variable dependencies.
  • A transformation to a local coordinate system whose axes are aligned to the so called main axes of the multidimensional distributions does not remove the original data relations – but there may exist a coordinate system in which the distribution data can be described in a simple, factorized form corresponding to a composition of seemingly un-correlated data distributions.

Anyway – by discussing density distributions we work on overall and large scale average relations between statistical value distributions for our variables, namely the z-vector components. We do not cover local micro-relations that may be in place in addition.

The relation of ellipses with Gaussian probability densities

Probability density functions for two logically separate, but maybe not un-correlated variables have to be multiplied. In our case this reflects the following point: First we determine the probability that the value of component x_i lies in a certain (infinitesimal) interval and then we determine the probability that (for the given value of x_i) the component x_j falls into another value range. The distributions for a specific variable can include variable relations and thus the probability density g(x_j) can include a dependency g(x_j(x_i)).

In the case of uncorrelated normal distributions per coordinate we can just multiply the individual Gaussians g(x_i) * g(x_j). Due to the quadratic terms in the exponent of the Gaussians we then get a sum of quadratic expressions in the common exponent, having the form fac1 * (x_i-mu_i)**2 + fac2 * (x_j-mu_j)**2.

By setting this expression to a constant value we get contour lines of the probability density distribution for the (x_i, x_j)-distribution. Quadratic sums correspond to the definition of an ellipse having main axes which are aligned with the x_i- and x_j-axes of the coordinate system. Thus the contour lines of a 2-dimensional distribution composed of un-correlated Gaussians are ellipses having an orientation aligned with the coordinate axes.

This was for un-correlated density-distributions of two vector components. Mathematically a linear correlation between a pair of Gaussians-distributions corresponds to an affine transformation of the contour-ellipses. The transformation can be expressed by a defined sequence of matrix operations describing a translation, rotations (in a defined order) and a dilation.

This means: The contour lines for a 2-dimensional probability density composed of linearly correlated Gaussians are still ellipses. But these ellipses will appear to be shifted, rotated and stretched along the main axes in comparison with their originally un-correlated Gaussian counterparts. The angle of rotation depends on details of the correlation function and the original standard deviations. The Pearson correlation matrix for linearly correlated distributions is a positive-definite one and, of course, shows off-diagonal elements different from zero. This result can be extended to multivariate normal distributions in spaces with many dimensions and related affine transformations of the coordinate system.

A multivariate normal distribution with linear correlations between the Gaussians results in elliptic contour lines for pair-wise density distributions in the respective 2D-coordinate planes of an orthogonal coordinate system. When we define the contours via multiples of the standard deviations of the underlying Gaussian functions we arrive at so called confidence ellipses.

A really nice mathematical aspect is that the basic parameters of the confidence ellipses can be derived from the normalized correlation coefficients of the Pearson matrix of the multivariate probability distribution. I will come back to this point in forthcoming posts in more detail. For now we just need to know that a multidimensional probability density comes along with confidence ellipses which can be calculated with the help of Pearson correlation coefficients.

Before we go on a word of caution: For a general multi-variate distribution it is not at all clear that it should decompose into a factorized form. However, for a multivariate normal distribution with un-correlated or only linearly correlated components this is by definition different. In this case a transformation to a coordinate system can be found which leads to a complete decomposition into a product of (seemingly) un-correlated Gaussians per component. The latter point lies at the center of PCA and SVD algorithms, which diagonalize the Pearson correlation matrix.

Do we really have Gaussian probability distributions for the individual z-vector-components?

After this short tour into the world of (multi-variate) normal distributions, Gaussian functions and related ellipses we are a bit better equipped to understand the density distributions in the latent space of our Autoencoders for human face images.

Let me remind you about the shapes of the number distribution for our concrete z-vector components resulting for for CelebA face images. The first plot shows the number densities on sampling intervals of width 0.25 for selected vector components resulting for case I of our experiments. The second plot shows the number densities for the values of selected components of case II.

Ok, these curves do resemble Gaussians and some fluctuations are normal. But can we prove the Gaussian properties of the curves a bit better?

Well, for case II I have drawn the best fits by Gaussian functions with the help of SciPy’s optimize.curve_fit() for 3 and yet another 4 selected components of the latent vectors and the respective number distribution curves. The dashed lines show the approximations by Gaussian functions:

The selected components are part of the list of around 20 dominant component distributions – due to their relatively large standard deviations. But the Gaussian form is consistently found for all components (with some small deviations regarding the symmetry of the curves).

So all in all it looks like as if our convolutional AE has indeed created a multivariate normal z-point distribution in the latent space. As said: This does not exclude correlations …

Pairwise linear correlations of the (normal) probability distributions for the latent vector components?

Now we are a bit bold – and assume the best case for us: Could the approxiate Gaussians distributions for the component values be pair-wise and linearly correlated? What would be a clear indication of a pair-wise linear correlation of our component distributions?

Well, we should find an elliptic form of contour lines in the 2-dimensional distribution for the component pair in the respective coordinate plane of the basic orthogonal LS coordinate system. This imposes quite strong symmetry conditions on the contour lines. The ellipses can be shifted and rotated – but they should remain being ellipses. If non-linear contributions to the correlation had a significant impact this would not be the case.

Practically it is not trivial to prove that we have approximately rotated ellipses in 2 dimensions. Satter plos alone do not help: Ellipses fit a lot of plotted distributions of discrete data points quite well. We really need to count number densities to get reliable contour lines. The following plots show such contour lines based on number sampling in rectangles and local smoothing operations with the help of scipy.stats.gaussian_kde().

The fat red and dark orange lines show corresponding confidence ellipses derived from the original CelebA distribution. See below for some remarks on confidence ellipses.

The contours are basically of elliptic shape although they do not show the complete symmetry expected for pure and linearly dependent Gaussian distributions. But overall the confidence ellipses fit quite well into the general form and orientation of the distributions. We also see that for higher σ-levels the coincidence with nearby contours is quite good. The wiggles in the contour change with the z-vector selection a bit.

We conclude that our basic impression regarding an elliptic shape of the z-point distributions is basically consistent with only linearly correlated Gaussian probability density distributions for the component values of the latent vectors.

Approximation of the core of the multivariate z-point distribution by confidence ellipses for component pairs

Above I referred to the boundary of a core of the probability density for two selected vector components. But how would we define the “boundary” of a continuous distribution in the coordinate planes? Answer: As we like – but based on the decline of the approximate Gaussian curves.

We can e.g. pick two times the half-width in each direction or we can use contours defined by confidence levels.
For 2 ≤ fact * σ ≤ 3 we saw already that the contour lines could well be fitted by confidence ellipses. A 3-sigma level covers around 97% of all data points or more. A 2 sigma-level ellipse encircles between 70% and 90% of all data points, depending on the eccentricity of the ellipse. Note that the numbers are smaller for ellipses than for rectangles. I.e. the standard 68-95-99.7 rule does not apply.

The plots below give you an impression of how well ellipses for a 2σ-confidence level approximate the core of the CelebA distribution in selected 2D coordinate planes of the latent space:

Each of the sub-plots was based on 10,000 statistically selected vectors of the 170,000 available in my test runs. This is a relative low number. Therefore, for a certain diameter of the points in the scatter plot only the inner core appears to be densely populated. The next plots shows the results for a 3 σ-level of the ellipse – but this time for 50,000 vectors. With more vectors we could visually fill the outer regions of the core.

The orange points mark the center of the multidimensional distributions derived from the one-dimensional distribution curves for the components. We see that it does not always appear to be optimally centered. There are multiple reasons: Our functions are not fully symmetric as ideal Gaussians. And equally important: The accuracy of the position depends on the sampling resolution which was coarse. Outliers of the distribution do have an impact.

And how would we explain the appearance of Gaussians and ellipses?

This all looks quite good, despite some notable deviations regarding symmetry and maxima. Gaussians fit at least most of the important probability density curves very well, though not by a 100%. The appearance of an elliptic shape of the inner core of the distribution and the appearance of overall elliptic contour curves can be explained by linear correlations of the Gaussian distributions for the components.

The appearance of normal distributions per component and basically linear correlations is something that really should be explained. I mean, dwell a bit on what we have found:

A convolutional Autoencoder network with more than 10 million adjustable parameters encoded information about human face images in the form of a roughly multivariate normal distribution of z-points in its latent space – with basically linear correlations between the Gaussian curves describing the probability densities functions for the component values of the z-vectors.

I find this astonishing and not at all self-evident. It is one of the most simple solutions for a multidimensional situation one can imagine. The following questions automatically came to my mind:

Does such a result only appear for training images of defined objects with some Gaussian variation in their features? Are the normal distributions a reflection of variations of relevant features in the original data?
Is this a typical result for (convolutional) AEs? How does it depend on the dimensionality of the latent space? Does it automatically come with a large number of z-space dimensions? Is it an efficient way to encode feature differences in the latent space, which (convolutional) AEs in general tend to use due to their structure?

Do I personally have a convincing explanation? No. Especially not, as the data shown above stem from convolutional neural networks [CNNs] without any batch-normalization layers.

A first idea would be that the dominant features of a human face themselves show variations described by Gaussian normal distributions already in the original data and that convolutional filtering does not destroy such distributions during optimization. A problem of this idea lies in the (non-) linear activation functions used at the nodes of the neural maps. Though ReLU, Leaky ReLU and SeLU contain linear parts.

The other problem is the linear form of the correlations. This is a rather simple kind of correlations. But why should an AE choose this simple form into its mapping of image information to latent space vectors after training?

How to generate statistical vectors for the creation of human face images?

The positive message which comes with the above results is that our problem of how to create proper statistical z-vectors decomposes into a sequence of two-dimensional problems. We can use the data of the ellipses appearing in the density-distributions for pairs of vector components to confine the components of statistically generated z-vectors to the relevant region in the latent space. All ellipses together restrict the component values in a well defined form. In the next post I will shortly outline some methods of how we can use the information contained in the ellipses with available algorithms.

Conclusion

In this post we have seen that for the case of a convolutional Autoencoder trained on CelebA human face images the latent vector distributions showed some remarkable properties:

The probability density functions for all component values can roughly be approximated by Gaussian functions. The components appear to be pairwise linearly correlated – at least to first order analysis. This automatically implies elliptic contour curves for the pairwise number density functions of coordinate values. Such contour curves were indeed found with first order accuracy. The core of the probability density for the z-points in the latent space could therefore be approximated by confidence ellipses for a σ-level above σ = 2.5.
The elliptic conditions correspond to a multivariate normal distribution with linear correlations of the variables.

Before we get to enthusiastic about these findings we should be careful and await a further test. All statements refer to a first order approximations. A real multivariate normal distribution would decompose into un-correlated Gaussians and 2D-ellipses of probability densities of component pairs after a PCA transformation.

In the next post

Autoencoders and latent space fragmentation – IX – PCA transformation of the z-point distribution for CelebA

I shall present the results of a PCA analysis. In later posts I will introduce a related method to restrict the components of statistical vectors to the relevant region in the latent space of our Autoencoder.

Links and literature

On first sight my short description of the relation between multivariate Gaussian normal distributions and ellipses as the contour lines for the projected density distributions on coordinate planes may appear plausible. But in the general multi-dimensional case the question of linear correlations requires some more math than indicated. For details I just refer to some articles on the Internet – but any good book on multivariate analysis will give you the relevant information
https://de.wikipedia.org/ wiki/ Multivariate_ Normalverteilung
https://de.wikipedia.org/ wiki/ Mehrdimensionale_ Normalverteilung
http:// www.mi.uni-koeln.de/ ~jeisenbe/ Vortrag2.pdf
https://methodenlehre.uni-mainz.de/ files/ 2019/06/ Multivariate-Distanz-Normalverteilung-MDC-Bayes.pdf
https://en.wikipedia.org/ wiki/ Multivariate_ normal_ distribution
https://en.wikipedia.org/ wiki/ Confidence_region
https://users.cs.utah.edu/ ~tch/ CS6640F2020/ resources/ How to draw a covariance error ellipse.pdf
https://biotoolbox.binghamton.edu/ Multivariate Methods/ Multivariate Tools and Background/ pdf files/ MTB%20070.pdf

Regarding the intimate relation between the ellipses’ main axes to normalized Pearson correlation coefficients I also refer to
https://carstenschelp.github.io/ 2018/09/14/ Plot_ Confidence_ Ellipse_ 001.html
I am very grateful that the author Carsten Schelp saved me a lot of time when trying to find a way to program a solution for confidence ellipses. Thank you, Mr. Schelp for the great work.